Sharp Morrey-Sobolev inequalities and eigenvalue problems on Riemannian-Finsler manifolds with nonnegative Ricci curvature

Kristály, Alexandru and Mester, Ágnes and Mezei, Ildiko (2022) Sharp Morrey-Sobolev inequalities and eigenvalue problems on Riemannian-Finsler manifolds with nonnegative Ricci curvature. Communications in Contemporary Mathematics. ISSN 0219-1997, ESSN: 1793-6683 (In Press)

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Abstract

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Krist\'aly [\textit{Math.\ Ann.}, in press, doi.org/10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey-Sobolev inequalities on $n$-dimensional Finsler manifolds having nonnegative $n$-Ricci curvature. A byproduct of this method is a Hardy-Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler-Laplace operator. Our results are also new in the Riemannian setting.

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